Reddit mentions of The Elements of Integration and Lebesgue Measure

An Introduction to Manifolds by Tu is a very approachable book that will get you up to Stokes. Might as well get the full version of Stokes on manifolds not just in analysis. From here you can go on to books by Ramanan, Michor, or Sharpe.

A Guide to Distribution Theory and Fourier Transforms by Strichartz was my introduction to Fourier analysis in undergrad. Probably helps to have some prior Fourier experience in a complex analysis or PDE course.

Bartle's Elements of Integration and Legesgue Measure is great for measure theory. Pretty short too.

Intro to Functional Analysis by Kreysig is an amazing introduction to functional analysis. Don't know why you'd learn it from any other book. Afterwards you can go on to functional books by Brezis, Lax, or Helemskii.

It depends on where they are and what the purpose is. If you are trying to discourage them (and there might be valid reasons to do that), I'd say try measure theory.

Maybe use the Bartle book.

That would give them a taste for how abstract things can get and also drive home the point tiny books can require a lot o work.

On the other hand, if you want to do something that will help them, they An Introduction to Mathematical Reasoning.

It won't break the bank and, despite a few small typos, covers a lot material fairly gently.

You've taken some sort of analysis course already? A lot of real analysis textbooks will cover Lebesgue integration to an extent.

Some good introductions to analysis that include content on Lebesgue integration:

Walter Rudin, principle of mathematical analysis, I think it is heavily focused on the real numbers, but a fantastic book to go through regardless. Introduces Lebesgue integration as of at least the 2nd edition (the Lebesgue theory seems to be for a more general space, not just real functions).

Rudin also has a more advanced book, Real and Complex Analysis, which I believe will cover Lebesgue integration, Fourier series and (obviously) covers complex analysis.

Carothers Real Analysis is the book I did my introductory real analysis course with. It does the typical content (metric spaces, compactness, connectedness, continuity, function spaces), it has a chapter on Fourier series, and a section (5 chapters) on Lebesgue integration.

Royden's real analysis I believe covers very similar topics and again has a long and detailed section on Lebesgue integration. No experience with it, recommended for my upcoming graduate analysis course.

Bartle, Elements of Integration is a full book on Lebesgue integration. Again, haven't read it yet, recommended for my upcoming course. It is supposed to be a classic on the topic from what I've heard.

My prof recommended me Bartle